Breaking trend panel unit root tests
Pui Sun Tam∗ March 2006
Abstract
This paper proposes Lagrange Multiplier based panel unit root tests allowing for structural breaks through simple extensions of existing group mean and combination tests. The proposed tests are more general than those previously suggested. They consider potential breaks in the intercept, in the slope, and both. A desirable property of the tests is their flexibility to accommodate heterogeneous break types across cross-sections in a panel. Response surfaces to approximate finite sample distributions of the underlying test statistics required to implement the panel tests are provided. The tests are analyzed for the case when the break dates are known and for the case when they are endogenously determined. A bootstrap test is further suggested to deal with cross-sectional dependency. The proposed tests are applied to two major macroeconomic variables, per capital gross domestic product and consumer prices of OECD countries.
JEL classification: C12, C15, C23
Keywords: Panel unit root, structural breaks, response surface, bootstrap
∗ Faculty of Business Administration, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau,
1.
Introduction
Notwithstanding their frequent use in applied works, most panel unit root tests in the literature ignore the presence of potential structural breaks. Erroneous omission of structural breaks may induce deceptive inference, as evidenced by Perron (1989) for the univariate case, and Murray and Papell (2000), Carrion-i-Silvestre et el. (2002),
Breitung and Candelon (2005) and Im et el. (2005, hereafter ILT) in the panel
setting. However, only a handful of studies, including Murray and Papell (2000), Carrion-i-Silvestre et el. (2002), Papell (2002), Tzavalis (2002), Bai and
Carrion-i-Silvestre (2004), Breitung and Candelon (2005) and ILT, have attempted to develop panel unit root tests with structural breaks. This paper addresses this concern and proposes extensions to the ILT test, and the combination tests of Maddala and Wu (1999) and Choi (2001) originally considered for the no break case to account for structural breaks.
The ILT test is noticeable among the prevailing panel unit root tests with structural breaks. It is a group mean test that combines individual exogenous intercept break tests, developed by Amsler and Lee (1995, hereafter AL), across cross-sections in a panel. Based on the Lagrange Multiplier (LM), or score, principle advanced by Schmidt and Phillips (1992, hereafter SP), the test allows for breaks under both the null and alternative hypotheses, so that rejection of the null unambiguously implies trend stationarity. It also permits a high degree of heterogeneity across cross-sections in a panel. However, two deficiencies of the test can be heeded. First, it only caters for intercept breaks, but for trending variables, it is possible for a break to affect the intercept of the trend function, as well as its slope. Erroneous determination of the break type may lead to incorrect conclusions (see Montañés et al., 2005). Second, it is
problem of pre-testing and data-mining (see Christiano, 1992). Thus, ILT apply the minimum LM test of Lee and Strazicich (2003) to select the break date for each individual, and then combine the minimum LM tests as in the known break case, but do not provide finite sample performance of the resulting panel unit root test.
This paper contributes in the following respects. First, we extend the ILT test to incorporate breaks in the intercept and slope, and pure breaks in the slope. Second, we show that the ILT test with minimum LM break selection exhibits severe over-rejections of the null. As a remedy, we apply the minimum sum of squared residuals (SSR) test of Nunes (2004) to select the break date for each time series. Notice that the ILT test, by combining test statistics, only has limited ability to permit heterogeneous break type across cross-sections in a panel. Thus, we also extend the combination tests to account for structural breaks. Since combination tests are based on combining p-values from unit root tests applied to individual time series in a panel,
they permit the use of a different unit root test on a different time series in the same panel. Consequently, combination tests permit full flexibility to accommodate heterogeneous break types across cross-sections.
We also simulate finite sample distributions of individual statistics required for computing the extended ILT and combination tests, and summarize the simulation results using response surfaces. We highlight in passing the importance of using finite sample distributions simulated under different lag order assumptions, which has not been thoroughly examined for panel unit root tests in the literature. We further suggest employing the bootstrap approach to compute the panel unit root tests when the cross-sectional independence assumption is relaxed.
The plan of this paper is as follows. Section Two details the testing methodology. Section Three deals with practical implementation issues of the proposed tests.
Section Four reports Monte Carlo simulation results, Section Five provides empirical applications, and Section Six concludes.
2. Testing Methodology
2.1 Model Framework
We consider the following general data generating process (DGP) for each cross-section in a panel
yit =µi+Zitδi+uit, uit =αiui,t−1+εit, εit~iidN(0,σi2), i=1,⋅ ⋅⋅,N, t=1,⋅ ⋅⋅,T, (1)
where Zit is the nonstochastic process excluding the intercept term, δi is the vector
of coefficients corresponding to Zit, ui0 is fixed, and iid means independently and
identical distributed. Thus, individual specific intercepts, time trend coefficients, structural break coefficients, serial correlation structure, first-order autoregressive parameters and break locations are allowed.1 No contemporaneous correlations among cross-sections are assumed, that is, εit is independent of εjs when i≠ j for
all t and s. The models of interest are specified as follows
(Model 0) Zit =t, δi =δ1i,
(Model 1) Zit=(t,DUit), δi=(δ1i,δ2i)′,
(Model 2) Zit=(t,DUit,DTit), δi=(δ1i,δ2i,δ3i)′,
(Model 3) Zit=(t,DTit), δi=(δ1i,δ3i)′,
where DUit=1 if t>TBi and 0 otherwise, DTit =t−TBi if t>TBi and 0 otherwise,
1 With serially correlated errors taking the general form, it
ε follows a stationary and invertible autoregressive moving average process Ai(L)εit=Bi(L)vit, where Ai(L) and Bi(L) are finite order polynomials in the lag operator L with all roots lying outside the unit circle and vit ~iidN(0,σvi2),
so that εit can be approximated by an autoregressive process of order ki,
it it i it i i L A L C L v B( )−1 ( )ε = ( )ε = , with
∑
= = ki j j j i L c L C 1 ) ( .i
TB (1<TBi <T) being the break date, also defined as TBi =λiT
(
0<λi <1)
, and λibeing the break fraction. Model 0 does not account for structural breaks. Model 1 allows for a change in the intercept of the trend function, Model 2 allows for a change in the intercept and slope, whereas Model 3 permits a change in the slope.
The null and alternative hypotheses are H0: αi =1, i=1,⋅ ⋅⋅,N,
H1: αi <1, i=1,⋅ ⋅⋅,N1, and αi=1, i=N1+1,⋅ ⋅⋅,N, (2)
where for test consistency, it is assumed N1/N is a constant as N→∞. Under the
null of αi =1, DGP (1) can be expressed as it i it it Z y =∆ δ +ε ∆ , t=2,⋅⋅⋅,T, yi1=µi+Zi1δi+ui0+εi1. (3)
Based on the LM, or score principle, SP suggest unit root testing for an individual using the regression equation
it k j ij it j i it t i i it S Z c S e y = +∆ + i ∆ + ∆
∑
= − −1 1 , , ~ ~ γ φ , t=ki+2,⋅⋅⋅,T . (4) where φi=αi−1, Sit yit µui Ztδi ~ ~ ~ − − = , t=2,⋅⋅⋅,T , with S~i1=0 , δi~ are the restricted
MLEs of δi obtained by a regression of the null equation (3), and µui yi µi Ziδi ~ ~
1
1− −
= ,
the restricted MLE of µui(≡µi+ui0). Equation (4) takes the augmented form to allow
for autocorrelated errors as in AL. The LM statistic is computed as the t-statistic for
testing φi=0. We denote the individual LM statistics under Models 0, 1, 2 and 3 as
0
iT
τ , τ1iT(λi), τiT2 (λi) and τiT3 (λi) respectively.
2.2 ILT Tests
to the SP distribution, with finite mean 0
i
E and variance Vi0 .
2 With the
cross-sectional independence assumption, the 0
iT
τ ’s are iid. Thus, the panel unit root
test under Model 0 is given by
0 0 0 ) ( i i NT V E N τ − , (5) where
∑
= = N i iT NT0 (1/N) 1τ0τ . The statistic (5) converges to a standard normal distribution as T and N grow large in sequence by the Lindeberg-Lévy central limit theorem (CLT)
under the null, and diverges to negative infinity under the alternative. ILT also consider Model 1 with exogenous known breaks. As shown by AL, under the null as
∞ →
T for fixed N, each τiT1 (λi) converges to the SP distribution, since ∆DUit in
equation (4) equals one at only one point and is thus asymptotically negligible. The panel unit root test under Model 1 with exogenous known breaks is
) ) ) ( ( 0 0 1 i i NT V E N τ λ − , (6) where ( ) (1/ ) ( ) 1 1 1 i N i iT NT λ N τ λ τ
∑
= = , with λ=(λ1,⋅ ⋅⋅,,λN).We extend the ILT testing methodology to Model 2 with intercept and slope breaks and Model 3 with slope breaks only. For Model 2, each τiT2 (λi) has a limiting
distribution that depends on the break location, with finite mean 2
i
E and variance Vi2,
under the null as T →∞ for fixed N, according to Lee and Strazicich (2003) and
Nunes (2004).3 Cross-sectional independence implies the τiT2(λi)’s are iid. Thus, we
2 The SP distribution is given by
[
1 2]
1/20 ( ) ) 2 / 1 ( ∫ − − Vi r dr , with Vi(r) Vi(r) Vi(r)dr 1 0 ∫ − = , a
demeaned Brownian bridge, Vi(r)=Wi(r)−rWi(1), a standard Brownian bridge, and Wi(r), a standard
Brownian motion.
3
The distribution is given by
[
1 2]
1/20 ( , )
2
1 −
∫
− Vi λi r dr , where Vi(λi,r) denotes the residuals
formulate the panel unit root test under Model 2 with exogenous known breaks as 2 2 2 2( ) [ ( ) ] i i NT V E N − = Γ λ τ λ , (7) where ( ) (1/ ) ( ) 1 2 2 i N i iT NT λ N τ λ τ
∑
== . For Model 3, under the null as T→∞ for fixed N,
each τiT3(λi) has a limiting distribution that depends on the break location, which is
the same as that for τiT2(λi), since inclusion of ∆DUit in equation (4) is asymptotically
irrelevant. Therefore, the panel unit root test under Model 3 with exogenous known breaks can be devised likewise as
3 3 3 3( ) [ ( ) ] i i NT V E N − = Γ λ τ λ , (8) where ( ) (1/ ) ( ) 1 3 3 i N i iT NT λ N τ λ τ
∑
== , Ei3 and Vi3 denote the mean and variance
respectively of the limiting distribution of τiT3 (λi).4
Another extension of the ILT test concerns the endogenous determination of break dates. ILT propose selecting the break date for each individual time series with the minimum break selection criterion, that is, the break date is selected where the LM statistic is minimized. Like the exogenous break test, the minimum LM test with an intercept break, denoted as τ1iT(λi(τ1iT)), converges in distribution to the SP test according to Nunes (2004). The panel unit root test under Model 1 using minimum LM break selection is calculated as in (6), where
∑
= = N i iT i iT T NT 1 N 1 1 1 1 (λ(τ )) (1/ ) τ (λ(τ )) τ , with )) ( , ), ( ( ) (τ1T λ1τ11T λN τ1NT λ = ⋅ ⋅⋅ , replaces τ1 (λ)
NT . However, the minimum LM test is
1 ) , ( r = duλi if r>λi and 0 otherwise, i i i i i i rW r W r V λ λ λ, ) ( ) ( ) ( = − if r≤λi, and [ (1) ( )] 1 ) ( ) ( ) ( 1 ) ( ) 1 ( ) ( ) ( ) ( ) , ( i i i i i i i i i i i i i i i i i i i i i i W W r W r W W W W r rW r W r V λ λ λ λ λ λ λ λ λ λ λ λ − − − − − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − − − − = if r>λi.
4 Since the limiting distributions of
) (
2
i iT λ
τ and τiT3 (λi) are the same, 3 2
i
i E
E = and Vi2=Vi3.
However, because finite sample distributions of the two statistics differ as shown in Section 3.1, we opt for the use of different notations under Models 2 and 3.
dominated by Nunes’ (2004) proposed minimum SSR test, where the break date is selected according to the estimator of TBi (orλi), that is, the date (or break fraction)
that minimizes the SSR in an ordinary least squares (OLS) regression of the null equation (3). The minimum SSR test with an intercept break, denoted as τi1(λˆi), converges in distribution to the SP test. This test is appealing since it selects the correct break date almost perfectly, and it possesses better finite sample properties than the minimum LM test. We calculate the panel unit root test under Model 1 using minimum SSR break selection by replacing τ1 (λ)
NT with =
∑
= N i iT i NT1 (λˆ) (1/N) 1τ1 (λˆ) τ , where λˆ=(λˆ1,⋅⋅⋅,λˆN), in (6).Application of the minimum LM and SSR break selection criteria to individual time series gives the test statistics τiT2(λi(τiT2)) and τiT2(λˆi) respectively under Model 2, and 3( ( 3)) iT i iT λ τ τ and 3(ˆ) i iT λ
τ respectively under Model 3. Since these criteria choose the
correct break date asymptotically as proved in Nunes (2004), limiting distributions of
)) ( ( 2 2 iT i iT λ τ τ and 2(ˆ) i iT λ
τ coincide with that of τiT2(λi), whereas limiting distributions of )) ( ( 3 3 iT i iT λ τ τ and 3(ˆ) i iT λ
τ are the same as τiT3(λi). Therefore, the panel unit root tests under
Models 2 and 3 using the two break selection criteria can be formulated likewise as in Model 1 through appropriate standardization of the simple averages of the corresponding individual test statistics.
2.3 Combination Tests
Maddala and Wu (1999) and Choi (2001) consider panel unit root testing under Model 0. Applying a one-sided left-tailed unit root test with no break to an individual time series, the asymptotic p-value of the resulting test statistic GiT, denoted as ∏i, is
∞ →
T . The four combination tests are the Fisher’s inverse chi-square test P, its
modified version Pm, the inverse normal test Z and the logit test L
∑
= ∏ − = N i i P 1ln( ) 2 . (9) ] ) 1 ) [ln( 1 1∑
= ∏ + − = N i i m N P . (10)∑
= − ∏ Φ = N i i N 1 1( ) 1 Z , (11)∑
= ⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ ∏ − ∏ = N i i i g 1ln 1 L , (12)where Φ(⋅) stands for a standard normal distribution function, and [3(5 +4)]
[
2 (5 +2)]
= N N N
g π . With cross-sectional independence, ∏i ~iidU[0,1] as T →∞
for fixed N under the null, so that −2ln(∏i)~iidχ2(2) , Φ−1(∏i)~iidN(0,1) and )]
1 ( [
ln ∏i −∏i have the logistic distribution with mean zero and variance π2 3.
Consequently, P~χ2(2N), Z~N(0,1), and approximately, L~t(5N+4), with t(⋅) being
the Student’s t-distribution function. Furthermore, as T and N grow large sequentially,
m P , Z and
∑
= ∏ −∏ ≈ N i i i N L 1 2 ) ln[ /(1 )] (3 π converge to a standard normal distribution by the CLT. Under the alternative, P and Pm diverge to positive infinity, whereas Z
and L diverge to negative infinity.
It is easy to see that the above testing procedure can readily be adapted to accommodate structural breaks. To this end, the DGP for each cross-section is first specified with the inclusion of appropriate structural break dummies, depending on the type of structural break. Next, the relevant unit root test with a break is applied to each time series, and the corresponding p-values obtained. These p-values are then
combined in accordance with (9) to (12) for panel unit root testing. It is noteworthy that because combination tests combine p-values, they are non-parametric in nature,
so that they allow for a different unit root test applied on a different individual time series with individual specific nonstochastic process, which can be specified under Model 0, 1, 2 or 3. Therefore, in effect, they permit full flexibility to accommodate heterogeneous break type across cross-sections in a panel. Such desirable property does not pertain to the ILT test, which combines individual statistics, so that its ability to accommodate heterogeneous break types is constrained by the limiting behavior of these individual statistics.
3. Practical Implementation Issues
3.1 Finite Sample Distribution
To implement the proposed tests, it is necessary to make available distributions of 0
iT τ , ) ( 2 i iT λ
τ and τiT3 (λi)and to obtain p-values, means and variances. Since the limiting
distributions are in general poor approximations in finite samples, we simulate finite sample distributions of the LM statistics for 31 sample sizes, with T = {30, 33, 35, 37, 40, 43, 45, 50, 55, 60, 65, 70, 75, 80, 90, 100, 125, 150, 175, 200, 225, 250, 275, 300, 350, 400, 450, 500, 750, 1000, 1250}. Nine known break fractions are considered, with λ = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} when a break is allowed. Since finite sample distributions also depend on the procedure employed to select k, two
ways of determining k are considered for each T under Model 0 and each combination
of T and λ under other models. The first way assumes k to be fixed at 0, 5 and 8,
whereas the second selects k by the general-to-specific t-sig method recommended by
Ng and Perron (1995), with the upper bound kmax set at 5 and 8.
The number of replications (I) used is as follows. When k = 0, 5, 8 and kmax = 5, I = 50,000, 30,000 and 20,000 for T≤ 100, 100 < T≤ 500, T > 500. When kmax = 8, I = 30,000, 20,000 and 10,000 for T≤ 100, 100 < T≤ 500, and T > 500. In each iteration,
T+100 observations are generated according to a driftless random walk with N(0,1)
errors while the initial value is set at zero, and the first 100 observations discarded to reduce the initial value effect.5 With the generated data, LM statistics are computed according to equation (4), and 399 equally spaced percentiles (from which p-values
can be obtained by linear interpolation as in Choi, 2001), means and variances of the statistics obtained and stored. All simulations are programmed in GAUSS 6.0.
Simulation results are summarized using estimated response surface regressions. Under Models 0 and 1, the response surface regression equation takes the form
ω β β β β β + + + + + = 0 1 1 2 31/2 3 12 4 13 ) ( T T T T T q , (13)
where q denotes the distribution quantiles, means or variances, and ω is the error term.
Under Models 2 and 3, the response surface regression equation is
ω β β λ λ λ λ λ = + + + + + + + 2 6 5 4 4 3 3 2 2 2 / 3 1 0( ) ( ) ( ) ( ) ( ) 1 1 ) , ( T T T g T g T g T g T g T q , (14) where ( ) 0 1 11/4 2 11/3 3 11/2 4 21/3 T T T T T gj =βj +βj +βj +βj +βj , j=0, 1, 2, 3, 4. It is worth
noting that under Model 1, the break fraction is found to be statistically insignificant in explaining the simulated quantiles, means and variances under all lag order assumptions. The parameters in (13) and (14) are estimated using OLS. However, since the errors are heteroskedastic, heteroskedasticity-consistent standard errors are computed using the jackknife covariance matrix estimator of MacKinnon and White (1985). The choice of response surface terms is motivated by the overall fit of the regression as measured by the coefficient of determination (R2), small standard error
of regression (σˆω), small mean absolute error (mean |ωˆ|), small maximum absolute error (max |ωˆ|), statistical significance of regressors and parsimony.
5 The individual exogenous break LM-type tests are exactly invariant to the break magnitudes
under the null when the break is correctly placed according to Nunes (2004). Furthermore, without loss of generality, the intercept and time trend coefficient are set to zero, and the error variance set to one.
Response surface regression results are reported in Tables 1 to 3. To conserve space, we only present results for k = 0, 5 and kmax = 5, and among the 399 pecentiles, the 5% quantile for illustration. With the reported estimates, the 5% quantiles, means and variances of the LM statistics for selected sample sizes and break fractions are computed and displayed in Table 4. Some interesting observations are revealed. First, response surface estimates provide good approximations to distributions of the LM statistics. Overall, the estimates track closely those simulated results available in the literature. Second, finite sample distribution of τ1iT(λi) does not differ significantly
from that of 0
iT
τ for any lag order choice. This suggests finite sample distribution of
0
iT
τ can also be used to compute the panel statistics under Model 1. In contrast, larger discrepancies are found between finite sample distributions of τiT2 (λi) and τiT2(λi), although they converge to the same limiting distribution. Third, finite sample distributions of the LM statistics under all models depend on the lag selection method employed. Using distributions of the LM statistics when k = 0 as benchmarks,
distributions of the statistics when k > 0 are shifted to the right, whereas distributions
of the statistics when kmax > 0 are shifted to the left. This draws attention to the use of appropriate distributions of the individual statistics in computing panel statistics under different lag order assumptions.
3.2 Cross-Sectional Dependency
The panel unit root tests discussed assume cross-sectional independence in a panel. Different methods for tackling cross-sectional dependency have been proposed in the literature. In this paper, we adopt the bootstrap approach as it can cater for cross-correlations of the general form
where ft is an unobserved stationary common factor, ϕi is the factor loading that
measures the heterogeneous impact of the common factor on each individual,
( )'
1t, , Nt
t υ υ
υ = ⋅ ⋅⋅ and Ωυ is not diagonal. Furthermore, in analyzing finite sample data,
bootstrap helps to obtain approximations to finite sample distributions of the panel statistics for more accurate statistical inference drawn.
The bootstrap procedure used is a recursive one based on resampling residuals from OLS estimation of the null model of each time series in a panel. Notice that computation of the panel statistics requires the use of distributions of the underlying individual statistics. Following conventional practice, finite sample distributions of individual statistics can be simulated as described in Section 3.1. An alternative method to obtain finite sample approximations is through the use of bootstrapped distribution of the individual statistic of interest, which is specific to the observed time series under study via a bootstrap procedure similar to that outlined above. Details of the bootstrap procedures are documented in the appendix.
4. Monte Carlo Simulation Results
We conduct Monte Carlo simulation experiments to assess finite sample performance of the panel unit root tests. Data are generated with N cross-sections and T+100 time
series observations for each cross-section according to DGP (1) with the following parameter values: µi=0, δ1i =0 δ2i=5, δ3i=1, ui0=0, and 1
2 =
i
σ for all i. The
break is set to occur at the middle of each time series. In analyzing size and power properties, αi=1 and αi=0.9 respectively for all i. The first 100 observations for
each cross-section are discarded to reduce the initial value effect.
Three experiments are conducted. In the first experiment, we analyze performance of the panel unit root tests under all models using different break
selection criteria while assuming no serial correlation and no cross-sectional dependency. Results are carried out for N = {10, 25, 50} and T = {25, 50}, with 3,000
iterations for each combination of N and T. In the second experiment, we introduce
serial correlation while breaks are assumed exogenously known so that the effects of using different lag orders are not obscured by the influence of break selection. The disturbances εit are generated by an AR(1) process εit=0.3εi,t−1+vit, vit~iidN(0,1). In
the third experiment, we consider cross-sectional dependency. The disturbances εit
are generated according to (15), with ϕi~U[−1,3] , which represents strong
cross-correlations. To ensure the covariance matrix Ωυ be symmetric, positive
definite and nonsingular, it is generated as in Chang (2002). Simulation results are obtained for N = {10, 25} and T = {25, 50}. In assessing the performance of the
bootstrap test, we use 500 Monte Carlo replications and 500 bootstrap replications. Distributions of individual LM statistics used in calculating the panel tests are simulated using the conventional approach as described in Section 3.3. Bootstrapped distributions of the LM statistics for use in the bootstrap test are simulated out of 5,000 replications. In computing the panel statistics under Model 1, distribution of
0
iT
τ is employed. All tests are conducted at the 5% nominal level. Size-adjusted powers are presented for fair comparison. All simulations are programmed in GAUSS 6.0. In what follows, we only report results for Models 1 and 2. Results for Models 0 and 3 are similar to those for Models 1 and 2 respectively.
In the first experiment, sizes of all tests are reasonably close to the 5% nominal level under the assumption of exogenous breaks. The inverse normal and logit tests exhibit most stable size and possess highest power among all tests. When break dates are endogenously determined with the minimum SSR criterion, the performance of all tests is comparable to that for the exogenous known break case. However, when break
dates are determined with the minimum LM criterion, all tests suffer from severe size distortions. The difference may be attributed to the higher frequency of true break date selection with the minimum SSR criterion. More importantly, finite sample distribution of τ1iT(λˆi) and τiT2(λˆi) are in line with that of τiT0 and τiT2(λi)
respectively. In contrast, finite sample distribution of τiT1 (λi(τiT1 )) and τiT2(λi(τiT2)) are
shifted to the left of 0
iT
τ and τiT2(λi).6 Consequently, selecting break dates by the
minimum LM criterion and then combining the resulting minimum LM test evidence across individuals to compute panel statistics using distribution of 0
iT
τ under Model 1 and distribution of τiT2(λi) under Model 2 is rendered inappropriate. Thus, we drop the
minimum LM break selection criterion in subsequent discussion.
In the second experiment with serial correlation, we consider both the fixed lag case and the sequential lag selection case. Results for the fixed lag case are displayed in Table 6. With AR(1) errors, the true lag orders are 1 for all i. When ki =0 for all i
are used, all sizes are reported to be zero. Size performance of the panel tests when using lag orders equal to the true ones is similar to that when using lag orders greater than the true ones. Under Model 1, sizes are in general not far from the nominal size of 5%, but combination tests (except the modified inverse chi-square test) have better size property than the ILT test overall. Under Model 2, all tests suffer from some size distortions when T is small relative to N, with the inverse chi-square test attaining the
best size property. For all model types, power decreases with higher lag order, with the inverse normal and logit tests exhibiting the highest power overall. Thus, with serial correlation, it is desirable to have T large relative to N, so that N/T does not
diverge as both N and T approach infinity simultaneously as mentioned in ILT.
6 The critical values of
)) ( ( 1 1 iT i iT λ τ
τ are substantially more negative than those of 0
iT
τ in finite samples according to simulation results in Nunes (2004) for the one-break case and Lee and Strazicich (2003) for the two-break case.
When the lag order is selected, we set the maximum lag order at 1/3
T . Two
methods of computing the panel tests are used. The first method entails first determining the optimal lag for all i and then calculating panel statistics using
distributions of individual statistics simulated with corresponding fixed lag order as in ILT. For the second method, panel statistics are calculated using distributions of individual statistics simulated with corresponding lag selection method. With the first method, all tests exhibit size distortions as shown in the upper panel of Table 7. ILT relate such kind of size distortions to the accuracy of true lag order selection. We attribute the size distortion to the use of inappropriate finite sample distribution of individual statistics, since finite sample distributions of individual statistics with endogenously determined lag order differ much from those with fixed lag order as shown in Section 3.1. This argument is evidenced by the fact that when distributions of individual statistics simulated with corresponding lag selection method are used to compute the panel statistics, the panel tests no longer suffer from severe over-rejections according to results in the lower panel of Table 7.
Data are cross-correlated in the third experiment. We find that panel tests taking no account of cross-sectional dependency suffer from size-distortions as shown in the upper panel of Table 8. Size biases accumulate as N grows, but are not attenuated by
increasing T. All panel tests become less powerful in the presence of cross-sectional
dependency. Results in the lower panel of Table 8 demonstrate the bootstrap approach helps to correct for size distortions of the panel tests in the presence of cross-correlations. The inverse normal test possesses the highest overall power.7
We draw the following conclusions from the simulation results. First, the inverse
7 Results shown in the lower panel of Table 8 are those using bootstrapped distributions of
individual statistics to compute the panel tests. Contrasting the use of conventional and bootstrapped distributions of individual statistics according to unreported results, size performance of the bootstrap test in either case is comparable, whereas power of the bootstrap test rise faster as T increases for fixed N using the latter.
normal and logit tests perform best in terms of size and power, and are therefore recommended for empirical applications. Second, in endogenizing the break dates instead of assuming exogenously known, the minimum SSR selection criterion is recommended for its high frequency of true break date selection and good size and power properties of the resulting panel tests. Third, bootstrap is an effective way to correct for size distortions even in the presence of strong cross-correlations which take on a general structure.
5. Empirical Applications
Since the seminal paper by Nelson and Plosser (1982), econometricians are increasingly interested in testing whether macroeconomic time series are integrated or trend stationary. We apply the proposed panel unit root tests using the minimum SSR break selection criterion to the study of two frequently investigated macroeconomic variables, per capita GDP and consumer prices, of OECD countries for the period 1960-2003 and 1949-2004 respectively.8 The logarithms of these series are analyzed. To carry out unit root testing with the panel data, individual LM regression for each country is first conducted with a 10% trimming at both ends of the time series. The optimal lag order is selected using the general-to-specific t-sig method with kmax = 8. The optimal break model type for each time series is selected following the procedure of Alba and Papell (1995). The most general model, Model 2, is first estimated. If both the estimated level and slope break coefficients are significant, Model 2 is applied. If only the slope coefficient is significant, Model 3 is used. If only the level coefficient is significant, Model 1 is employed. If both coefficients are not
8 Data for per capita GDP measured in PPP for 25 OECD countries are obtained from the OECD
Health Data 2005. These GDP data are expressed in constant 2000 prices using GDP deflators taken from the World Development Indicators 2005. Data for consumer prices for 23 OECD countries, with 2000 as the base year, come from the International Financial Statistics April 2005.
significant, the no break situation is considered. Test results for each country are presented in the upper panel of Table 9. It can be noted that Model 0 is rarely selected for individual time series, thus justifying the allowance for structural breaks in unit root testing. Furthermore, Models 2 and 3 are more frequently selected than Model 1, which signifies the importance of accounting for a break in the slope. Comparing the computed individual statistics with the bootstrapped critical values obtained according to the procedure outlined in the appendix using 5,000 replications, it is found that the unit root null for per capita GDP is rejected for three countries and that for consumer prices for five countries, at conventional significance levels.
The null hypothesis of cross-sectional independence is further tested with the Pesaran (2004) test. The test statistic is
(
∑ ∑
−)
= =+ − 1 1 1ˆ ) 1 ( 2 N i N i j ij N N T ϑ , where ij ϑˆ is the
sample estimate of the pairwise correlation of the residuals, which follows a standard normal limiting distribution. The computed test statistics are 15.666 for per capita GDP, and 21.454 for consumer prices, thus we reject the null of cross-sectional independence at conventional significance levels for both variables.
In the presence of cross-sectional correlations, the bootstrap approach as described in Section 3.2 is employed. The bootstrapped critical values of the panel statistics are obtained out of 5,000 replications. Panel unit root test results using bootstrapped distributions of individual statistics are presented in the lower panel of Table 9.9,10 The joint unit root null for per capita GDP is not rejected, thus supporting the non-stationary nature of the variable. As for consumer prices, results provide evidence of trend-stationarity. The joint unit root null is rejected at the 5% level using
9 Results using conventional distributions, are qualitative the same and are therefore not reported. 10 For the group mean test, when asymptotic critical values are used for hypothesis testing, its
flexibility in accommodating heterogeneous break type across cross-sections is restricted as discussed in Section 2.2. Accommodating full flexibility like the combination tests will result in nonstandard distribution, the critical values of which can be obtained using bootstrap.
the inverse chi-square test and its modified version, and at the 1% level using other panel tests.
6. Conclusion
This chapter proposes LM-type group mean and combination tests for panel unit root testing with structural breaks. The proposed tests have the following advantages. First, they are modeled with breaks under both the null and alternative hypotheses. Second, they are able to account for breaks in the intercept, in the slope and both. Third, they take into account the heterogeneity nature of cross-sections in a panel by allowing for individual specific intercepts, time trend and structural break coefficients, serial correlation structure, first-order autoregressive parameters, break locations and break types. Simulation results show that two of the combination tests, namely the inverse normal and logit tests, have the best size and power properties. Furthermore, in endogenizing the break date, the minimum SSR criterion is preferred to the more commonly employed minimum LM criterion. Finally, bootstrap is shown to be effective in correcting for size distortions arising from a relaxation of the cross-sectional independence assumption. The proposed panel unit root tests with structural breaks are applied to study the time series behavior of per capita GDP and consumer prices for OECD countries. Empirical results lend support to the nonstationary nature of the former and the trend stationary behavior of the latter.
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Appendix
A. Bootstrapping critical values for panel unit root test
(a1) Select the null model for each individual time series in the observed panel data according to null equation (3) with the incorporation of serial correlation
it k j ij it j i it it Z c y v y
∑
i = ∆ − + + ∆ = ∆ 1 δ . (A1.1)The optimal ki is determined by the general-to-specific t-sig method. Where a
break is allowed, the optimal ki is first determined at each possible break point,
then all possible regressions with the optimal ki estimated, and the break date
selected where the Schwarz Bayesian Criterion is minimized.11 (a2) Estimate the selected null model, defined at the selected break date TBi
∧
and the corresponding optimal kˆi, for every individual with the observed data by OLS
and obtain the parameter estimatesδˆi, cˆi1, …,
i
k i
cˆˆ together with the regression
residuals vˆit, t=kˆi+2,⋅ ⋅⋅,T . The residuals are then rescaled by the multiplicative
factor li /(li −oi), where li is the number of observations used in estimation
and oi is the number of parameters estimated, to obtain v(it, t=kˆi+2,⋅ ⋅⋅,T .
(a3) Construct RP number of pseudo panel samples. For each pseudo panel sample, resamplev(t=(v(1t,⋅⋅⋅,v(Nt)randomly with replacement T times to obtainvt**=
(
v1*t*,⋅⋅⋅,v*Nt*)
,T
t=1,⋅ ⋅⋅, .12 For each individual time series, construct the first-differenced series
* * ˆ 1 * * * * ˆ ˆ it k j ij it j i it it Z c y v y
∑
i = ∆ − + + ∆ = ∆ δ . (A1.2)The initial starting values for all individuals together
(
**)
0 * * ˆ, ,y y−k⋅ ⋅⋅ , with ˆ max(ˆi) i k k= and(
** **)
1 * * , , Nj j j y yy = ⋅ ⋅⋅ , j=−kˆ,⋅ ⋅⋅,0 , are obtained by block resampling. The
observed data are divided into (T −kˆ) overlapping blocks each with (kˆ+1) time
observations per individual, and one block is randomly selected with replacement to serve as the initial values. Partial sums of **
it
y
∆ give the level series
∑
= ∆ + = t j j i it y y y 1 * * * * 0 * * . (A1.3)(a4) Calculate the bootstrapped individual statistic of interest according to equation (4) for each time series in each pseudo panel sample. Construct the bootstrapped panel statistic of interest with these bootstrapped individual statistics for each pseudo panel sample. The p-values, means and variances used to compute the
panel statistic are obtained in two ways. The first is using conventional distributions of the individual statistics simulated as described in Section 3.1. The second is using bootstrapped distributions of the individual statistics simulated in steps (b4) and (b5) below. [In step (b5), “observed individual statistic” is replaced by bootstrapped individual statistic of interest in this step.] A sorted vector of these RP bootstrapped panel statistics forms the bootstrapped distribution of the panel statistic under the null, from which bootstrapped critical values of the panel statistic are obtained.
11 The estimated break date for each individual here is the estimate under the null, conditional on
the selected null model. This may differ from the break date generating the statistic for unit root testing.
12 Resampling the residuals as a vector for fixed
t for all individuals simultaneously help to preserve
B. Bootstrapping critical values, mean, variance and p-vale for individual unit root test
(b1) Same as step (a1). (b2) Same as step (a2).
(b3) Construct RU number of pseudo samples for each individual time series. First, resample v(it, t=kˆi+2,⋅ ⋅⋅,T , randomly with replacement T times to obtain v*it,
T
t=1,⋅ ⋅⋅, . Then construct the first-differenced series *
it y ∆ * ˆ 1 * * ˆ ˆ it k j ij it j i it it Z c y v y
∑
i = ∆ − + + ∆ = ∆ δ . (A2.1)The initial starting values ⎜⎝⎛
( )
− ⋅ ⋅⋅ * ⎟⎠⎞0 * ˆ , , i k i y y
i are obtained by random draw with
replacement one block from the (T−kˆi) overlapping blocks of the observed individual time series, each with length (kˆi +1). Partial sums of ∆yit* gives the
level series
∑
= ∆ + = t j j i it y y y 1 * * 0 * . (A2.2)(b4) Calculate the bootstrapped individual statistic of interest according to equation (4) for each pseudo sample. A sorted vector of these RU bootstrapped statistics forms the bootstrapped distribution of the individual statistic under the null, from which bootstrapped critical values, mean and variance can be computed. (b5) Obtain p-value for the individual statistic as #{bootstrapped individual statistic
in step (b4)≤observed individual statistic}/RU, where observed individual statistic is the test statistic calculated according to equation (4) using the observed data yit.
Table 1. Response surface estimates under Models 0 and 1. Coefficients & statistics 5% quantile Mean Variance 5% quantile Mean Variance 5% quantile Mean Variance Model 0 k = 0 k = 5 kmax = 5 β0 -3.0175 -1.9663 0.3328 -3.0152 -1.9702 0.3323 -3.0220 -1.9711 0.3323 β1 -5.8713 -1.4399 1.2707 -12.007 1.2539 -1.6262 -27.319 -9.5769 2.9874 β2 244.2308 54.9976 47.1178 84.6323 45.4104 96.9871 β3 206.8227 79.463 -1189.27 -263.008 -182.136 -114.944 -228.564 -650.862 β4 -4387.95 -1650.10 254.337 6922.293 3006.236 3810.814 -457.827 2112.922 7034.919 R2 0.9426 0.8550 0.9666 0.7731 0.997 0.9964 0.9967 0.9977 0.9978 ω σˆ 0.0088 0.0020 0.0029 0.0089 0.0033 0.0030 0.0099 0.0034 0.0033 Mean |ωˆ| 0.0065 0.0015 0.0023 0.0063 0.0024 0.0021 0.0069 0.0022 0.0025 Max |ωˆ| 0.0166 0.0040 0.0051 0.0189 0.0071 0.0071 0.0239 0.0080 0.0066 Model 1 k = 0 k = 5 kmax = 5 β0 -3.0250 -1.972 0.3290 -3.0262 -1.9727 0.3324 -3.0067 -1.9648 0.3280 β1 -3.3665 -0.2814 2.1107 4.3835 -1.8594 -37.3495 -1..3793 10.7862 β2 1.9455 53.9962 196.9223 46.6379 -39.0829 β3 -15.2599 -4.9768 -26.1806 60.5926 28.4646 -226.803 -353.439 -115.870 β4 711.7967 -3149.09 910.8902 4806.765 -3937.18 3370.686 R2 0.9572 0.7169 0.978 0.7745 0.997 0.9964 0.9977 0.9987 0.9978 ω σˆ 0.0080 0.0027 0.0026 0.0078 0.0032 0.0030 0.0076 0.0023 0.0030 Mean |ωˆ| 0.0064 0.0021 0.0020 0.0062 0.0025 0.0023 0.0058 0.0017 0.0024 Max |ωˆ| 0.0228 0.0057 0.0073 0.0199 0.0076 0.0103 0.0324 0.0075 0.0092
Notes: k is the fixed lag order used in regression. kmax is the upper bound of the lag order used when
the lag order is determined endogenously in regression. R2 is the coefficient of determination. σˆω is
the estimated standard error of the regression. Mean |ωˆ| gives the mean absolute error of the response surface predictions versus simulated values, whereas Max |ωˆ| denotes the maximum absolute error.
Table 2. Response surface estimates under Model 2. Coefficients & statistics 5% quantile Mean Variance 5% quantile Mean Variance 5% quantile Mean Variance k = 0 k = 5 kmax = 5 β00 -3.3650 -1.1379 1.7187 27.3195 25.8458 -2.0058 -3.0710 9.7186 9.4432 β01 44.4158 -30.5227 -76.6409 -1295.14 -1272.73 31.214 46.4865 -528.316 -444.315 β02 -115.328 58.2877 171.8811 2666.469 2697.047 0.6939 -135.189 1112.130 966.3929 β03 153.4493 -46.8127 -189.566 -2454.34 -2657.94 -160.030 211.5373 -1077.17 -1011.46 β04 -91.3441 19.9652 102.3988 1089.328 1281.366 174.8088 -144.674 505.0783 522.7624 β10 -144.416 9.8848 96.2237 5043.148 1864.997 -2516.03 1364.446 1108.409 -440.979 β11 194.9303 -50.8515 -144.274 -14748.7 -4530.52 7664.678 -4580.11 -2909.38 1863.976 β12 -123.979 39.2899 94.9235 12566.42 3607.005 -6620.57 4051.674 2385.562 -1737.80 β13 8.6147 -3.9079 -7.1657 -2028.96 -501.342 1099.268 -701.415 -355.360 328.4013 β14 163.7555 33.901 -91.3299 60.3871 26.2779 -30.394 β20 267.6613 -67.0487 -206.017 -11189.3 -3636.31 5652.536 -3376.03 -2382.51 1200.230 β21 -352.534 176.0439 304.5694 32671.04 8508.458 -17211.4 11134.86 6137.928 -4800.21 β22 221.271 -129.740 -200.065 -27837.2 -6672.37 14867.01 -9820.72 -5007.69 4428.554 β23 -14.8551 12.0473 15.0541 4496.330 892.3936 -2469.15 1692.846 738.219 -823.765 β24 -363.185 -57.0283 205.2491 -145.346 -53.9591 75.3743 β30 -222.633 108.3534 200.8407 10628.67 2914.369 -5338.17 3599.857 2177.166 -1331.23 β31 289.0128 -230.955 -294.495 -30989.2 -6385.68 16268.12 -11742.6 -5466.67 5176.935 β32 -179.135 8 166.3468 193.5095 26401.66 4863.656 -14054.4 10339.79 4425.527 -4748.78 β33 11.6137 -14.8942 -14.5764 -4264.66 -599.826 2334.945 -1778.57 -641.362 875.5189 β34 344.5736 33.3175 -194.162 152.5529 45.9403 -79.5811 β40 102.5449 -50.5124 -92.5182 -4486.28 -1116.54 2194.693 -1608.60 -894.804 581.519 β41 -136.764 104.7302 136.7118 13083.22 2330.573 -6699.70 5254.857 2214.726 -2270.93 β42 85.4693 -75.2058 -90.0777 -11146.6 -1732.32 5789.019 -4630.08 -1783.52 2084.434 β43 -5.6627 6.7000 6.8265 1800.4 198.0079 -961.858 797.4862 255.2959 -384.635 β44 -145.447 -9.3079 79.9713 -68.4952 -17.9992 34.9807 R2 0.9948 0.9985 0.9876 0.9836 0.9973 0.9742 0.9958 0.9986 0.9824 ω σˆ 0.0126 0.0076 0.0045 0.0295 0.0123 0.0153 0.0193 0.0098 0.0122 Mean |ωˆ| 0.0091 0.0047 0.0034 0.0211 0.0085 0.0102 0.0145 0.0064 0.0081 Max |ωˆ| 0.0406 0.0330 0.0131 0.0919 0.0488 0.0683 0.0608 0.0406 0.0556
Table 3. Response surface estimates under Model 3. Coefficients & statistics 5% quantile Mean Variance 5% quantile Mean Variance 5% quantile Mean Variance k = 0 k = 5 kmax = 5 β00 -3.6489 -1.0263 1.5889 20.6507 24.2763 0.7546 -7.3648 8.8846 10.9748 β01 54.1158 -34.6824 -70.2494 -1009.39 -1200.64 -82.297 231.4024 -491.442 -505.236 β02 -134.429 66.3181 158.0253 2075.780 2543.678 230.2551 -518.086 1034.938 1087.692 β03 170.8513 -52.9362 -175.257 -1904.81 -2504.96 -360.121 569.1555 -1003.16 -1112.59 β04 -100.053 21.8433 95.0905 843.6152 1206.332 254.358 -305.243 470.6864 559.7532 β10 -98.8721 20.6314 83.7165 3964.255 1625.891 -2005.91 940.451 928.9196 -169.342 β11 110.7526 -71.2776 -120.512 -11608.5 -3857.91 6204.270 -3378.63 -2381.14 1075.579 β12 -66.3714 53.4469 78.4884 9889.661 3039.158 -5381.95 3037.506 1933.028 -1066.47 β13 3.939 -5.0916 -5.7999 -1596.05 -411.060 900.8314 -540.825 -281.311 220.0619 β14 128.7458 26.7041 -75.418 47.6957 20.2049 -21.6496 β20 160.6153 -83.0126 -172.289 -8738.81 -3047.66 4561.337 -2422.73 -1913.30 645.5827 β21 -153.626 207.2012 241.1837 25516.64 6852.940 -14077.5 8411.401 4757.615 5 -3180.23 β22 84.8515 -151.480 -156.322 -21731.1 -5274.11 12206.33 -7514.88 -3824.61 3046.567 β23 -3.7291 13.8861 11.4296 3506.088 669.7494 -2042.02 1325.277 544.2838 -599.969 β24 -282.858 -39.2396 170.9282 -116.076 -38.0125 57.2519 β30 -104.676 110.5854 154.7544 8164.847 2249.812 -4339.52 2626.101 1609.825 -867.474 β31 68.4929 -237.668 -209.027 -23777.9 -4523.50 13389.79 -8925.23 -3802.48 3806.860 β32 -27.5136 171.4427 134.6985 20240.56 3291.837 -11608.3 7943.570 2999.463 -3576.47 β33 -0.8227 -15.3833 -9.7235 -3263.09 -349.683 1941.576 -1392.84 -407.503 684.6781 β34 263.0989 13.3254 -162.504 121.5016 26.6848 -64.0576 β40 42.1844 -48.6371 -65.9632 -3394.06 -798.256 1784.613 -1167.34 -615.333 408.474 β41 -23.8674 102.6427 87.8002 9885.146 1442.354 -5516.85 3965.465 1396.757 -1754.30 β42 7.7864 -74.0234 -56.4739 -8413.63 -983.449 4783.890 -3529.84 -1082.90 1641.357 β43 0.7234 6.6388 4.0601 1355.890 79.0682 -800.303 619.1879 140.462 -312.262 β44 -109.263 0.1825 66.9814 -54.0416 -8.5451 29.0811 R2 0.9947 0.9985 0.9892 0.9826 0.9976 0.9773 0.9966 0.9987 0.9869 ω σˆ 0.0123 0.0074 0.0041 0.0292 0.0113 0.0137 0.0168 0.0090 0.0103 Mean |ωˆ| 0.0088 0.0047 0.0032 0.0212 0.0078 0.0095 0.0123 0.0057 0.0071 Max |ωˆ| 0.0427 0.0320 0.0137 0.0836 0.0518 0.0578 0.0553 0.0356 0.0479
Table 4. Comparison of response surface estimates and simulated results in the literature.
T Lag order λ Measure Simulated results in literature Response surface estimates
Source Value Model 0 Model 1 Model 2 Model 3
25/50/1,000 k=0 5% quantile SPa (Model 0) -3.18/-3.11/-3.02 -3.202/-3.087/-3.023 -3.184/-3.098/-3.028
38 k=0 mean/variance ILTb (Model 0) -1.979/0.371 -1.979/0.371 -1.983/0.375
38 k=5 mean/variance ILT (Model 0) -1.829/0.430 -1.830/0.434 -1.821/0.444
56 k=0 mean/variance ILT (Model 0) -1.976/0.358 -1.976/0.357 -1.979/0.360
56 k=5 mean/variance ILT (Model 0) -1.884/0.377 -1.883/0.379 -1.880/0.383
62/111 k=0 5% quantile ALc (Model 1) -3.09/-3.06 -3.077/-3.057 -3.083/-3.055
1,000 k=0 0.3/0.5/0.8 5% quantile Nunesd (Models 2, 3) -3.59/-3.66/-3.48 -3.598/-3.658/-3.490 -3.596/-3.658/-3.491
38 k=0/5 5% quantile -3.109/-2.986 -3.124/-2.986 38 kmax=5 5% quantile -3.480 -3.456 38 kmax=5 mean/variance -2.149/0.502 -2.145/0.507 38 k=0 0.3/0.5/0.8 5% quantile -3.751/-3.828/-3.603 -3.739/-3.826/-3.638 38 k=5 0.3/0.5/0.8 5% quantile -3.338/-3.840/-3.423 -3.333/-3.801/-3.484 38 kmax=5 0.3/0.5/0.8 5% quantile -4.121/-4.342/-4.027 -4.098/-4.311/-4.073 38 k=0 0.5 mean/variance -2.680/0.393 -2.678/0.390 38 k=5 0.5 mean/variance -2.586/0.532 -2.572/0.382 38 kmax=5 0.5 mean/variance -3.054/0.528 -3.030/0.519
Notes: a SP denotes Schmidt and Phillips (1992). b ILT denotes Im, Lee and Tieslau (2005), where effective number of observations is used instead of actual number of
Table 5. Sizes and size-adjusted powers of panel unit root tests under Models 1 and 2.
N T Γ P Z L Pm
Size Power Size Power Size Power Size Power Size Power
Model 1 Exogenous 10 25 0.059 0.103 0.053 0.097 0.050 0.115 0.053 0.108 0.069 0.097 50 0.055 0.378 0.046 0.304 0.047 0.429 0.045 0.418 0.059 0.304 25 25 0.057 0.155 0.054 0.135 0.050 0.179 0.051 0.182 0.061 0.135 50 0.054 0.730 0.043 0.620 0.048 0.765 0.048 0.752 0.051 0.620 50 25 0.052 0.271 0.049 0.237 0.042 0.308 0.046 0.304 0.057 0.237 50 0.056 0.943 0.049 0.862 0.046 0.968 0.047 0.965 0.056 0.862 Minimum LM 10 25 0.641 0.094 0.556 0.096 0.623 0.099 0.612 0.098 0.606 0.096 50 0.593 0.270 0.528 0.229 0.563 0.294 0.559 0.269 0.576 0.229 25 25 0.926 0.127 0.883 0.120 0.927 0.134 0.925 0.134 0.898 0.120 50 0.895 0.500 0.854 0.441 0.876 0.557 0.879 0.523 0.874 0.441 50 25 0.997 0.196 0.991 0.183 0.998 0.213 0.997 0.200 0.992 0.183 50 0.993 0.816 0.988 0.724 0.992 0.856 0.992 0.822 0.989 0.724 Minimum SSR 10 25 0.059 0.100 0.053 0.090 0.051 0.108 0.052 0.105 0.069 0.090 50 0.053 0.391 0.045 0.315 0.045 0.437 0.044 0.423 0.058 0.315 25 25 0.058 0.166 0.054 0.141 0.050 0.193 0.051 0.194 0.061 0.141 50 0.055 0.712 0.041 0.619 0.048 0.762 0.048 0.750 0.050 0.619 50 25 0.053 0.259 0.049 0.215 0.044 0.294 0.047 0.286 0.058 0.215 50 0.056 0.931 0.048 0.849 0.045 0.961 0.046 0.960 0.054 0.849 Model 2 Exogenous 10 25 0.055 0.082 0.049 0.075 0.046 0.085 0.045 0.084 0.063 0.075 50 0.055 0.169 0.048 0.148 0.049 0.177 0.046 0.187 0.059 0.148 25 25 0.059 0.078 0.054 0.079 0.058 0.082 0.057 0.082 0.064 0.079 50 0.058 0.282 0.049 0.245 0.050 0.307 0.051 0.298 0.060 0.245 50 25 0.055 0.105 0.049 0.099 0.049 0.118 0.053 0.106 0.055 0.099 50 0.057 0.463 0.044 0.382 0.046 0.492 0.045 0.498 0.052 0.382 Minimum LM 10 25 0.896 0.062 0.806 0.066 0.917 0.066 0.903 0.062 0.847 0.066 50 0.932 0.127 0.867 0.127 0.935 0.135 0.925 0.127 0.893 0.127 25 25 0.999 0.068 0.993 0.071 1.000 0.066 1.000 0.068 0.996 0.071 50 1.000 0.191 0.998 0.177 1.000 0.207 1.000 0.187 0.999 0.177 50 25 1.000 0.091 1.000 0.092 1.000 0.090 1.000 0.088 1.000 0.092 50 1.000 0.295 1.000 0.257 1.000 0.320 1.000 0.293 1.000 0.257 Minimum SSR 10 25 0.055 0.081 0.049 0.074 0.046 0.084 0.045 0.082 0.063 0.074 50 0.055 0.165 0.048 0.150 0.049 0.177 0.046 0.188 0.059 0.150 25 25 0.059 0.081 0.053 0.082 0.057 0.083 0.056 0.083 0.064 0.082 50 0.060 0.281 0.051 0.243 0.050 0.305 0.050 0.298 0.062 0.243 50 25 0.056 0.102 0.049 0.101 0.050 0.114 0.052 0.108 0.055 0.101 50 0.059 0.443 0.045 0.382 0.047 0.481 0.046 0.490 0.052 0.382
Notes:Γ is the group mean test following Im, Lee and Tieslau (2005), whereas P, Z, L and Pm are the
combination tests, namely the inverse chi-square, inverse normal, logit and modified inverse chi-square tests, respectively, following Maddala and Wu (1999) and Choi (2001).
Table 6. Sizes and size-adjusted powers of panel unit root tests under Models 1 and 2 (exogenous breaks) with serial correlation, fixed lag in regression.
N T Γ P Z L Pm
Size Power Size Power Size Power Size Power Size Power
Model 1 k = 0 10 25 0.000 0.112 0.000 0.097 0.000 0.114 0.000 0.123 0.000 0.097 50 0.000 0.379 0.000 0.276 0.000 0.444 0.000 0.460 0.000 0.276 25 25 0.000 0.166 0.000 0.138 0.000 0.188 0.000 0.199 0.000 0.138 50 0.000 0.727 0.000 0.557 0.000 0.787 0.000 0.803 0.000 0.557 50 25 0.000 0.273 0.000 0.209 0.000 0.321 0.000 0.336 0.000 0.209 50 0.000 0.945 0.000 0.804 0.000 0.974 0.000 0.979 0.000 0.804 k = 1 10 25 0.052 0.104 0.046 0.094 0.047 0.112 0.046 0.111 0.059 0.094 50 0.060 0.286 0.047 0.248 0.050 0.311 0.051 0.293 0.062 0.248 25 25 0.063 0.141 0.060 0.122 0.054 0.152 0.057 0.145 0.067 0.122 50 0.063 0.570 0.051 0.471 0.047 0.624 0.051 0.607 0.062 0.471 50 25 0.057 0.184 0.057 0.152 0.057 0.194 0.055 0.197 0.065 0.152 50 0.067 0.819 0.051 0.717 0.057 0.867 0.060 0.859 0.058 0.717 k = 2 10 25 0.061 0.081 0.054 0.084 0.058 0.074 0.058 0.077 0.069 0.084 50 0.062 0.249 0.056 0.191 0.053 0.265 0.054 0.252 0.069 0.191 25 25 0.065 0.106 0.062 0.091 0.060 0.113 0.063 0.106 0.073 0.091 50 0.062 0.469 0.053 0.379 0.052 0.514 0.052 0.506 0.064 0.379 50 25 0.073 0.126 0.071 0.114 0.066 0.127 0.068 0.130 0.080 0.114 50 0.063 0.743 0.049 0.639 0.051 0.783 0.051 0.776 0.054 0.639 Model 2 k = 0 10 25 0.000 0.074 0.000 0.065 0.000 0.070 0.000 0.066 0.000 0.065 50 0.000 0.165 0.000 0.125 0.000 0.163 0.000 0.162 0.000 0.125 25 25 0.000 0.078 0.000 0.072 0.000 0.074 0.000 0.081 0.000 0.072 50 0.000 0.273 0.000 0.196 0.000 0.295 0.000 0.300 0.000 0.196 50 25 0.000 0.126 0.000 0.094 0.000 0.133 0.000 0.128 0.000 0.094 50 0.000 0.466 0.000 0.318 0.000 0.506 0.000 0.518 0.000 0.318 k = 1 10 25 0.096 0.067 0.078 0.063 0.095 0.065 0.092 0.063 0.094 0.063 50 0.076 0.106 0.063 0.100 0.070 0.114 0.070 0.106 0.081 0.100 25 25 0.148 0.060 0.114 0.059 0.158 0.062 0.156 0.061 0.131 0.059 50 0.091 0.172 0.069 0.163 0.087 0.169 0.088 0.171 0.082 0.163 50 25 0.219 0.076 0.160 0.073 0.252 0.083 0.239 0.081 0.174 0.073 50 0.116 0.255 0.085 0.211 0.115 0.273 0.114 0.258 0.094 0.211 k = 2 10 25 0.118 0.051 0.093 0.054 0.110 0.053 0.108 0.054 0.121 0.054 50 0.082 0.092 0.065 0.090 0.078 0.100 0.072 0.097 0.081 0.090 25 25 0.179 0.045 0.153 0.040 0.176 0.047 0.175 0.046 0.171 0.040 50 0.093 0.142 0.070 0.133 0.093 0.140 0.090 0.137 0.083 0.133 50 25 0.264 0.057 0.217 0.059 0.266 0.060 0.256 0.057 0.235 0.059 50 0.135 0.193 0.092 0.174 0.135 0.206 0.134 0.205 0.103 0.174
Table 7. Sizes and size-adjusted powers of panel unit root tests under Models 1 and 2 (exogenous breaks) with serial correlation, lag selection in regression.
N T Γ P Z L Pm
Size Power Size Power Size Power Size Power Size Power
Distribution of individual statistic with fixed lag
Model 1 10 25 0.174 0.076 0.170 0.069 0.133 0.074 0.138 0.074 0.201 0.069 50 0.127 0.212 0.117 0.186 0.096 0.243 0.105 0.222 0.142 0.186 25 25 0.253 0.121 0.264 0.111 0.188 0.118 0.202 0.122 0.289 0.111 50 0.169 0.416 0.167 0.340 0.127 0.453 0.139 0.432 0.189 0.340 50 25 0.365 0.128 0.389 0.114 0.274 0.138 0.294 0.132 0.412 0.114 50 0.211 0.656 0.224 0.576 0.157 0.705 0.164 0.685 0.240 0.576 Model 2 10 25 0.567 0.067 0.497 0.064 0.560 0.059 0.550 0.063 0.544 0.064 50 0.319 0.078 0.275 0.078 0.301 0.076 0.296 0.077 0.319 0.078 25 25 0.892 0.078 0.841 0.070 0.897 0.074 0.894 0.075 0.865 0.070 50 0.567 0.101 0.497 0.103 0.555 0.098 0.554 0.097 0.533 0.103 50 25 0.995 0.038 0.987 0.036 0.995 0.030 0.995 0.033 0.989 0.036 50 0.827 0.141 0.763 0.126 0.827 0.140 0.821 0.130 0.781 0.126
Distribution of individual statistic with lag selection
Model 1 10 25 0.031 0.077 0.031 0.070 0.021 0.078 0.024 0.084 0.038 0.070 50 0.038 0.211 0.032 0.182 0.028 0.249 0.028 0.226 0.043 0.182 25 25 0.010 0.122 0.016 0.115 0.009 0.124 0.008 0.124 0.020 0.115 50 0.025 0.420 0.023 0.354 0.017 0.452 0.018 0.451 0.028 0.354 50 25 0.009 0.132 0.012 0.117 0.007 0.143 0.007 0.133 0.015 0.117 50 0.018 0.669 0.015 0.576 0.009 0.723 0.009 0.709 0.019 0.576 Model 2 10 25 0.066 0.044 0.058 0.046 0.059 0.044 0.059 0.044 0.074 0.046 50 0.045 0.089 0.041 0.088 0.042 0.089 0.040 0.088 0.051 0.088 25 25 0.069 0.047 0.071 0.041 0.055 0.049 0.058 0.051 0.082 0.041 50 0.038 0.120 0.035 0.119 0.039 0.118 0.038 0.118 0.043 0.119 50 25 0.070 0.055 0.071 0.058 0.055 0.054 0.057 0.051 0.078 0.058 50 0.031 0.178 0.025 0.160 0.031 0.193 0.031 0.178 0.033 0.160
Table 8. Sizes and size-adjusted powers of panel unit root tests under Models 1 and 2 with cross-sectional dependence.
N T Γ P Z L Pm
Size Power Size Power Size Power Size Power Size Power
No account for cross-sectional dependency
Model 1 Exogenous 10 25 0.166 0.088 0.133 0.088 0.176 0.089 0.166 0.088 0.148 0.088 50 0.170 0.200 0.131 0.187 0.177 0.218 0.165 0.208 0.147 0.187 25 25 0.266 0.088 0.218 0.083 0.293 0.095 0.284 0.091 0.227 0.083 50 0.273 0.211 0.218 0.189 0.297 0.231 0.289 0.217 0.228 0.189 Minimum SSR 10 25 0.162 0.081 0.127 0.078 0.163 0.089 0.156 0.085 0.143 0.078 50 0.162 0.183 0.123 0.171 0.166 0.196 0.159 0.185 0.140 0.171 25 25 0.258 0.083 0.209 0.079 0.280 0.087 0.273 0.084 0.218 0.079 50 0.266 0.193 0.213 0.178 0.285 0.209 0.278 0.198 0.223 0.178 Model 2 Exogenous 10 25 0.180 0.065 0.141 0.061 0.191 0.064 0.180 0.065 0.159 0.061 50 0.178 0.115 0.134 0.102 0.182 0.117 0.170 0.116 0.152 0.102 25 25 0.272 0.058 0.226 0.058 0.301 0.060 0.291 0.058 0.236 0.058 50 0.273 0.101 0.214 0.093 0.292 0.105 0.279 0.102 0.225 0.093 Minimum SSR 10 25 0.142 0.063 0.114 0.059 0.142 0.063 0.137 0.062 0.132 0.059 50 0.144 0.099 0.111 0.095 0.142 0.105 0.135 0.102 0.125 0.095 25 25 0.230 0.065 0.195 0.062 0.246 0.061 0.235 0.064 0.204 0.062 50 0.230 0.090 0.192 0.087 0.239 0.095 0.229 0.090 0.202 0.087
Account for cross-sectional dependency using bootstrap
Model 1 Exogenous 10 25 0.046 0.093 0.052 0.098 0.048 0.092 0.050 0.088 0.052 0.098 50 0.038 0.170 0.038 0.162 0.036 0.184 0.038 0.172 0.038 0.162 25 25 0.047 0.081 0.052 0.092 0.042 0.076 0.046 0.078 0.052 0.092 50 0.039 0.175 0.042 0.166 0.040 0.196 0.038 0.182 0.042 0.166 Minimum SSR 10 25 0.059 0.100 0.058 0.098 0.060 0.102 0.058 0.098 0.058 0.098 50 0.049 0.191 0.048 0.170 0.048 0.204 0.050 0.206 0.048 0.170 25 25 0.062 0.115 0.066 0.128 0.064 0.114 0.062 0.120 0.066 0.128 50 0.054 0.212 0.052 0.190 0.052 0.220 0.054 0.210 0.052 0.190 Model 2 Exogenous 10 25 0.061 0.070 0.064 0.076 0.054 0.064 0.056 0.066 0.064 0.076 50 0.036 0.107 0.042 0.098 0.038 0.112 0.038 0.106 0.042 0.098 25 25 0.048 0.058 0.048 0.060 0.048 0.060 0.046 0.058 0.048 0.060 50 0.047 0.111 0.052 0.112 0.046 0.118 0.048 0.110 0.052 0.112 Minimum SSR 10 25 0.062 0.068 0.062 0.068 0.058 0.074 0.062 0.064 0.062 0.068 50 0.048 0.111 0.050 0.104 0.044 0.112 0.044 0.112 0.050 0.104 25 25 0.056 0.062 0.058 0.068 0.058 0.066 0.060 0.062 0.058 0.068 50 0.054 0.123 0.054 0.116 0.054 0.130 0.056 0.124 0.054 0.116
